How to make sense with Syllogism

How to make sense with Syllogism?

BL Burney MS 275, f. 184r (check foliation) Here a female figure, a personification of the liberal art of logic/dialectic, holds a branch which flowers into the ‘syllogism’, a cornerstone of Aristotelian logic. The figures on the right – teacher and pupil – are depicted conversation. This scene, heavy with gestural components, conveys a sense of the orality of medieval teaching. However, rules of syllogistic reasoning where also learned through books. What did these books look like?

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Reasoning through syllogism

Propositions and syllogisms are the meat and two veg of medieval logical reasoning. Derived from the writings of Aristotle – notably the De interpretatione [logica vetus] and the Prior analytics [logica nova] – in the translations of Apuleius and Boethius, propositions* and syllogisms* provided a methodology for constructing a logical argument.

Tekst verschijnt wanneer je met je muis op het woord gaat staan:

Proposition:

a subject and predicate linked with a verb (cupola)

For example: ‘All horses are animals’, where ‘horses’ is the subject, ‘animals’ is the predicate (something which says something about the subject) and ‘are’ is the verb which links both parts.

Syllogism:

A type of argument consisting of two premises (a type of proposition) which lead to a conclusion

Propositions vary in ‘quantity’ (being universal, particular, indefinite or singular) and in ‘quality’ (affirmative or negative)

‘All horses are animals’, for example, is a universal affirmative proposition. 

‘Some horses are not animals’, is a particular, negative proposition (and false!)

In order to relate propositions to each other and use them to generate arguments, a device called a ‘square of opposition’ was used.

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How does a square of opposition work? 

I: The propositions

In this example, the universal affirmative proposition is ‘all men are just’ [see text in the red box]

It’s placed on the top of the square alongside the other universal, this time negative, proposition ‘no men are just’ [see text in yellow box]

At the bottom of the square we find the particular affirmative and negative propositions: ‘some men are just’ and ‘some men are not just’ [see text in green box]

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II: Contradictory relationships

The diagonal lines crossing the centre of the diagram (labelled ‘contradictorie’) connect propositions which are contradictory to each other. For example, if ‘all men are just’ (the universal affirmative proposition) then ‘some men are not just’ (the particular negative proposition) is a contradictory statement. If ‘all men are just’ then the statement ‘some men are not just’ cannot be true.

Similarly, reading across the other diagonal, if ‘no men are just’ then ‘some men are just’ cannot be true.

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III: Contrary relationships

The arrangement of the universal affirmative and negative propositions at the top of the square and the particular affirmative and negative propositions at the bottom is also meaningful; these are described as being ‘contrary’ to each other. 

That is, it is not possible for both of the universal propositions – ‘all men are just’ and ‘all men are not just’ – to be true, although they could be both false.

However, while it is possible for both particular propositions – ‘some men are just’ and ‘some men are not just’ – to be true,  they cannot both be false.

[these are described as ‘subcontrary’, link to BPL 88, f. 48r which has these contrary and subcontrary relationships indicated]

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IV: Subaltern relationships

Finally, reading alongside the side of the square, a further relationship between the propositions, that of subalternation, is identified.

If the universal affirmative proposition ‘all men are just’ is true, then the particular affirmative proposition ‘some man are just’ must also be true.

Similarly, if the universal negative proposition, ‘no man is just’ is true, then the particular negative proposition, ‘some man is not just’ must also be true.

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How do propositions relate to syllogisms?

The square of opposition allows you to relate propositions that have the same subject and predicate (here ‘man’, ‘being just’). 

But what happens when you have more than two terms, or want to relate propositions with different subjects or predicates? A syllogism is another way of relating propositions and formally consists of two propositions and a conclusion.

One term is common to both propositions, or premise, and to the conclusion. By reasoning from what is common between the premises you can come to a reasoned conclusion:

For example: 

Every horse is an animal.

Some horses are white.

Therefore, some animals are white.

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Moods of syllogisms

Aristotle identified nineteen different ‘moods’ or patterns of syllogisms, which he divided into three ‘figures’, or groupings, with each figure involving a different arrangement of subject and predicate in the premise.

In this manuscript (BPL 84, f. 63r, a copy of Boethius’ work Introductio ad syllogismos categoricos - CHECK), we can see an exposition of the these three figures. They are numbered alongside the text in the inner margin and red brackets and rubrics are used to distinguish the three figures.

E.g. Figure 1.1

‘All that is just is good. All that is virtuous is just. Therefore, all that is virtuous is good.’

This syllogism joins two universal affirmative propositions (‘all that is just is good’ and ‘all that is virtuous is just’) and deduces the conclusion ‘all that is virtuous is good’.

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In order to remember the nineteen moods of syllogisms, medieval logicians came up with a short verse which ‘encoded’ the details of each of the three figures:

Barbara Celarent Darii Ferio Baralipton

Celantes Dabitis Fapesmo Friesomorum

Cesare Cambestres Festino Barocho Darapti

Felapto Disamis Datisi Bocardo Ferison

To understand how this worked let us look at our first example of the first figure:

‘All that is just is good. All that is virtuous is just. Therefore, all that is virtuous is good.’

This syllogism consists of two universal affirmative premises and a universal affirmative conclusion.

This syllogism is represented by the word ‘Barbara’, where the three vowels ‘a, a, a’ indicate that this is a syllogism made up for three universal affirmatives (represented by ‘a’, taken from ‘affirmo’)

The vowel ‘e’ represents a universal negative (taken from ‘nego’), while i and o represent the particular affirmative and the particular negative respectively (with i taken from ‘affirmo’ and o from ‘nego’)

An influential account of the moods of the syllogisms, popularizing this verse is found in Peter of Spain’s Summulae. In this manuscript, BnF lat. 16611, given to the Sorbonne by the thirteenth-century master, Gerard de Abbeville, the names of the syllogisms have been added in the space between the two columns of text.

‘<Barbara> [erased], Celarent, Darii, Ferio, Baralipton, Celantes’

Here these names do double-duty; they remind the reader of the mnemonic verse, but they also indicate where the description of each mood can be found in the text.

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Playing with syllogisms I: Leiden UB, BPL 88

This bifolium (f. 1 – 2) was attached to the ninth-century portion of BPL 88 as a set of flyleaves. We know it became appended to the Martianus text in the eleventh century at the latest, as on f. 2v an eleventh-century hand has added an ‘accessus’, a short introductory text, adding detail about Martianus and his writings, suggesting that by this point the bifolium and text were joined. [suggest add detail about accessus to Mariken’s slides on BPL 88, link to student practices from theme – also detail about offset on f. 1r?]

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On f. 1v-2r a scribe has added a series of ‘squares of oppositions’ and syllogisms; at first these look like the scribe is experimenting with the possibilities of the ‘square of opposition’. It’s tempting to think of this as an exercise sheet – perhaps evidence of a student trying out various options! 

However, when we look closely at the content we find that the collection of schemes is far from random. All of these sets of propositions can be found in Boethius’ first commentary on his translation of Aristotle’s  De interpretatione (= Peri Hermeneias). Compare BPL 25 where these sets of propositions (with the exception of the last) are found in the margins. The bifolium is, effectively, a diagrammatic summary of De interpretatione!

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The scribe has gathered all these sets of propositions into one place. Was he a student, making a medieval ‘crib sheet’, looking to memorise all the propositions from the text, perhaps wanting to have all the schemes to hand in one place as he worked his way through Boethius’ complex text? 

Another possibility is that this bifolium could be a draft sheet for a scribe; the fact that the second scheme on f. 1v duplicates and corrects the content of the first suggests that this sheet could represent ‘work in progress’. The unfinished zodiacal scheme on f. 2r might lead us to the same conclusion. Schemes were not easy to plan and draw and draft sheets must have been made, although they rarely survive.

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Playing with syllogisms II: Leiden UB, BPL 139 (Paris BnF lat. 6638)

This manuscript contains a ‘wheel of syllogisms’, an alternate means of representing the relationship between propositions than the ’square of opposition’.

The wheel is found at the conclusion of a copy of Apuleius’ (c. 123 – 170) exposition on Aristotle’s Peri hermenias. The wheel represents a portion of Apuleius’ text.

‘It is clear that someone who has proposed something asserts to it. But either universal is destroyed in three ways: when its particular is shown to be false or when either of the two others – its contrary or contradictory is shown to be true. However it is established in one way: if its contradictory is shown to be false. On the other hand a particular proposition is destroyed in one way: if its contradictory is shown to be true. However it is established in three ways: if its universal is true or if either of the two others – its subcontrary or contradictory is false’. [translation from S. Gersh, Concord in Discourse..., p. 168-9]

‘Certum est enim, quod concedat, qui aliquid proposuerit. destruitur autem utravis universalis trifarium, dum aut particularis eius falsa ostenditur aut utravis ex duabus ceteris versa, sive incongrua, sive subneutra. instruitur autem uno modo, si alterutra eius falsa ostenditur. contra particularis, uno quidem modo destruitur, si alterutra eius vera ostenditur; instruitur autem trifarium, si aut universalis eius vera est aut utrauis ex duabus ceteris falsa, sive subpar eius sive subneutra.’ [Londry et al, p. 88]

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Here ‘A’ represents the ‘universal affirmative’, ‘B’ the ‘universal negative’, ‘C’ the ‘particular affirmative’ and ‘D’ the ‘particular negative’. 

Each quadrant represents how to deal with one type of proposition : here the highlighted section in yellow deals with the ‘universal affirmative’ (indicated as ‘A in the central section, here in red). The text in the outer ring of the circle reads ‘contra universalem affirmationem’, ‘against the universal affirmative’.

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The segments of the quadrant offer the four possibilities:

‘D: Si falsa est astruitur; C: Si falsum est destruitur; B: Si verum est destruitur; D: Si verum est destruitur’. That is, if D, the particular negative, is false the universal affirmative is true; if C, the particular affirmative, is false the universal affirmative is false; if B, the universal negative, is true, the universal affirmative is false; if D, the particular negative, is true, the universal affirmative is false.

The wheel shows that scholars experimented with different ways to represent relationships between propositions aside from the ‘square of opposition’.

This manuscript was copied at Fleury (the next codicological unit contains some of the logical works of Abbo of Fleury). We know Abbo used schemes in his works, but how was this one used in practice?

A clue can be found in another copy of the wheel, found in BnF lat. 6638. This manuscript [link to it] contains a fragment of De syllogismis hypotheticis of Abbo of Fleury (f. 1va-4rb), followed by Apuleius’ work. Here the scheme is added on a small piece of parchment bound into the manuscript. Unlike the copy in BPL 139 B, this copy could have been rotated [can we make this dynamic somehow?], allowing the reader to follow the propositions around the wheel more easily. The fact that it was copied on a small sheet suggests that this might have been a study aid, a loose piece of parchment to have to hand while reading Apuleius’ text.

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Playing with syllogisms III: Leiden UB, VLF 48 [link]

Here two scribes have used the blank space at the end of this manuscript of Martianus Capella’s De Nuptiis as a space for writing some notes: this shows how ‘unused’ space could become a venue in which material only tangentially related to the content of the codex was gathered.

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The first scheme is a ‘proof’ of the utility of dialectic. Here dialectical methods are harnessed to prove that the art itself has merit, but the ‘proof’ also provides an example of dialectical argumentation in action.[see yellow arrow] The inspiration for the scheme is Martianus’ text itself. In De nuptiis, IV.422, the figure of Dialectic offers the following example of an argument which combines a categorical and a conditional syllogism: ’if the question is whether dialectic itself is advantageous, the argument should be set out: if it is advantageous to discuss well, the science of discussing well is advantageous; but to discuss well is advantageous; therefore dialectic is advantageous.’ (trans. Stahl, p. 152 = Si quaestio sit, utrum utilis sit ipsa dialectica proponendum est: si bene disputare utile est, utilie est bene disputandi scientia; at bene disputare utile est, utilis est igitur dialctica’. )

The scheme is evidence of a reader trying to understand the steps made by Dialectic in reaching this conclusion; the scheme illustrates the reader’s thought process!

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Text of first logical scheme (reveal when floating over the text? How best to combine text + translation? 

Predicatiuus: Categorical:

Utrum utilis sit dialectica Whether dialectic is useful

Omnis disciplina utilis est All disciplines are useful

Dialecta autem disciplina est Dialectic is also a discipline

Utilis est igitur dialectica Therefore dialectic is useful

Conditionalis: Conditional:

Utrum utilis sit disciplina scientia Whether the discipline of knowledge is useful

Si bene disputatare utilis est, utilis est bene disputandi scientia If discussing well is useful,

 the knowledge of discussing well  is useful.

At bene disputare utile est But good discussion is useful

Utilis est igitur bene disputandi scientia Therefore the knowledge of discussing well is useful 

Mixtus. A coniusgationes [sic] Mixed. By combining.

Utrum utilis est disputandi scientia Whether the knowledge of discussing is useful

Si bene disputare utile est, utile est bene disputandi scientia If good discussion is useful, 

the science of discussing well is useful

At bene disputare utilis est But good discussion is useful

Utilis est igitur dialectia. Therefore dialectic is useful.

In the second set of schemes [see blue arrow] lines connect terms of the propositions and conclusion in order to indicate how the argument is constructed. Visualising the argument in this way helps understand why it is valid. In the first set of propositions, the lines show how the predicate of the first premise becomes the subject of the next, with the terms then joined together in the conclusion. In the second set (which the scribe introduces with the words ‘uel alio modo’, or in another manner)  the subject of the first premise becomes the predicate of the next. Again lines are used to show how terms shift from proposition to conclusion.

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Omne iustum honestum Every just thing is honourable

Omne honestum pulc<h>rum Every honourable thing is beautiful

Omne igitur iustum pulc<h>rum Therefore every just thing is beautiful

Omne iustum honestum Every just thing is honourable

Omne bonum iustum Every good thing is just

Omne igitur bonum honestum Therefore every good thing is honourable.

Playing with syllogisms IV: Leiden UB, BPL 84

This segment of text from Boethius’ Introductio ad syllogismos categoricos, an introduction to Aristotle’s theory of syllogisms, gives an example of how one ‘converts’ a proposition. The first line reads:

‘Omnis homo rationalis est’ converts to ‘Nullus homo non rationalis est’

‘Every man is rational’ converts to ‘No man is not-rational’

This is an example of ‘conversion by obversion’ (if every A is B, then no A is ‘non-B’).

Conversion was a useful technique for the medieval logician as it allowed new propositions to be generated from existing ones. 

Although this is quite a technical discussion, the scribe has still found an opportunity to make his reader laugh, making the initial ‘Q’ out of a man and woman making love. Her blush and guilty look at the reader gives the impression that the couple have been caught in the act!

Perhaps the reader would have also got a laugh out of this list of commonplaces, or ‘loci’, used for argumentation found in another collection of Boethian logical works. Here the list is decorated with a flourish that on closer inspection is clearly male genitals!  LINK

These sexual details have not been defaced or erased, suggesting that the (most likely religious) reader of the texts was not without a sense of humour.

An elaborate scheme has been added at the end of the Introductio ad syllogismos categoricos, offering a summary of the characteristics of ‘simple propositions’. The scheme combines two parts of Boethius’ text; first his explanation of simple propositions (PL 64, 769C-769D) and then some of the examples he gives later in the text of how such propositions work and convert.

Here the scheme offers three examples of simple propositions which share one term (either subject or predicate)

‘homo animal est. animal substantia est.’ [Man is an animal; an animal is a substance’]

‘homo grammaticus est. homo musicus est.’ [Man is a grammarian; man is a musician’]

‘homo animal est. equus animal est.’ [Man is an animal. A horse is an animal]

Here the scheme represents Boethius’ account of ‘conversion by opposition’ (‘contraposition’)

The second set of propositions: ‘Omnis non homo animal est. Omne animal non homo est.’ (’All non-man is an animal; all animals are not man’) shows how switching the order of the subject and predicate of the sentence generates a new proposition. DOUBLE CHECK

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The compiler of this scheme was interested in the theoretical make-up of logical argumentation. Although its primary function is to summarise the different forms which simple propositions could take, following Boethius’s text, it could also have been used to generate new syllogisms and test new examples, with students perhaps substituting new terms for the samples of subject and predicate given.